Evaluate the Summation sum from i=1 to 6 of 4i-2

Problem

(∑_i=1^6)(4*i−2)

Solution

1. Identify the summation as a finite series with 6 terms where the general term is (a_i)=4*i−2

2. Apply the linearity property of summations to split the expression into two separate sums.

(∑_i=1^6)(4*i−2)=4*(∑_i=1^6)(i)−(∑_i=1^6)(2)

3. Use the summation formula for the first n integers, (∑_i=1^n)(i)=(n*(n+1))/2 where n=6

(∑_i=1^6)(i)=(6*(6+1))/2

(∑_i=1^6)(i)=42/2=21

4. Evaluate the sum of the constant term, where (∑_i=1^n)(c)=n*c

(∑_i=1^6)(2)=6×2=12

5. Substitute the values back into the linear expression and simplify.

4*(21)−12=84−12

84−12=72

Final Answer

(∑_i=1^6)(4*i−2)=72

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